Question

identify the identity: (1-sin theta) / (1+sin theta) = (tan theta -sec theta)^2

Answer 1

If you don't mind, I'll forgo the writing of \(\theta\) because it's a bit tedious.
\[\begin{align*}\frac{1-\sin}{1+\sin}&=\frac{(1-\sin)(1-\sin)}{(1+\sin)(1-\sin)}\\
&=\frac{(1-\sin)^2}{1-\sin^2}\\
&=\frac{1-2\sin+\sin^2}{\cos^2}\\
&=\frac{1}{\cos^2}-2\frac{\sin}{\cos^2}+\frac{\sin^2}{\cos^2}\\
&=\sec^2-2\tan\sec+\tan^2\\
&=(\sec-\tan)^2\\
&=((-1)(\tan-\sec))^2\\
&=(-1)^2(\tan-\sec)^2\\
&=(\tan-\sec)^2\end{align*}\]